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Prove that: inttan^n x\ dx=1/(n-1)tan^("...

Prove that: `inttan^n x\ dx=1/(n-1)tan^("n"-1)x-inttan^(n-2)x\ dx`

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To prove the recurrence relation for the integral of \( \tan^n x \), we need to show that: \[ \int \tan^n x \, dx = \frac{1}{n-1} \tan^{n-1} x - \int \tan^{n-2} x \, dx \] ### Step 1: Start with the integral Let \( I_n = \int \tan^n x \, dx \). ...
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Knowledge Check

  • inttan^(2)x" " dx=

    A
    `tan x+x+c`
    B
    `-tanx-x+c`
    C
    `-tan x+x+c`
    D
    `tanx-x+c`

  • inttan(sin^(-1)x)dx=

    A
    `1/sqrt(1-x^(2))+c`
    B
    `sqrt(1-x^(2))+c`
    C
    `x/sqrt(1-x^(2))+c`
    D
    `-sqrt(1-x^(2))+c`

  • inttan^(2)((x)/(2))dx=

    A
    `(1)/(2)tan((x)/(2))+c`
    B
    `2tan((x)/(2))-x+c`
    C
    `sec((x)/(2))+c`
    D
    `2sec((x)/(2))+c`

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